Pulsating flow in a heat exchanger

The problem of heat transfer in industrial processes, heat exchangers, and combustion chambers is formulated for a case where flow inside the chamber consists of a periodic motion imposed on a fully developed turbulent flow. It is shown that the velocity pulsations induce harmonic oscillations in temperature, thus breaking the temperature field into a steady mean part and a harmonic part. The interaction between the velocity and temperature oscillations introduces an extra term into the energy equation which reflects the effect of pulsations in producing higher heat transfer rates. The analysis shows that when the mean temperature is fully developed with constant heat flux at the wall, there is no effect of the velocity pulsations on the total heat transfer rate along the chamber. For the case where the mean temperature profile is not fully developed, analytical solutions are obtained for asymptotic values of the pulsations frequency. The results show the temperature gradient and its dependence on the frequency. These results are used to evaluate the feasibility of pulsating the flow in a heat exchanger for obtaining higher rates of heat transfer.     

Pulsating turbulent pipe flow in the current dominated regime at high and very-high frequencies

The paper presents Direct Numerical Simulations of sinusoidal pulsating turbulent flow, at low bulk Reynolds numbers, with high frequency, in a straight pipe. Our objective is to study pulsating flow considering it as the superposition of a temporal unsteadiness on a mean current, and from this viewpoint, to decompose the flow in a mean and an oscillating part. Firstly, we examine the time-averaged statistics, which show that the parent flow retains its properties. Then, we analyze the oscillating part of the flow, and confirm the notion that for rapidly pulsating flow, the amplitude of the streamwise velocity and the phase lag at different radial locations follow the solution of the laminar Stokes problem. In addition, we find that the modulation of the turbulent fluctuations follows approximately the sinusoidal form of the imposed pulsation, and that the ratio of the frequency parameter to the amplitude of the streamwise velocity can be used as a scaling factor. We investigate the effects of the amplitude and the frequency of the imposed unsteadiness on the modulation of the time-averaged properties and the turbulence statistics, through a systematic analysis. Finally, we examine the time evolution of the mean velocity and the turbulent fluctuations. These results indicate that a lower limit for the high frequency regime can be identified, based on the level of conformity of the phase-averaged profiles on their steady-state counterparts. For very high frequencies, we find that that the flow behavior does not change, indicating the absence of an upper limit for the high frequency regime.     

Ritz method for oscillatory flow in ducts

Oscillatory viscous flow is solved by a Ritz integration method. The method is robust and efficient. The elliptic duct and the isosceles triangular duct are studied in detail. We find that the unsteady flow is dependent on a non‐dimensional frequency s and the aspect ratio b. In general, for s=O(1) or lower, the flow may be considered quasi‐steady, and the velocity is in phase with the applied pressure gradient. For large frequency, s=O(100), the maximum velocity occurs near the larger curvature ends and corners. Copyright © 2010 John Wiley & Sons, Ltd.     

Solution of magnetohydrodynamic flow in a rectangular duct by Chebyshev collocation method

In this study, matrix representation of the Chebyshev collocation method for partial differential equation has been represented and applied to solve magnetohydrodynamic (MHD) flow equations in a rectangular duct in the presence of transverse external oblique magnetic field. Numerical solution of velocity and induced magnetic field is obtained for steady‐state, fully developed, incompressible flow for a conducting fluid inside the duct. The Chebyshev collocation method is used with a reasonable number of collocations points, which gives accurate numerical solutions of the MHD flow problem. The results for velocity and induced magnetic field are visualized in terms of graphics for values of Hartmann number H≤1000. Copyright © 2010 John Wiley & Sons, Ltd.     

Effects of Reynolds number on physiological‐type pulsatile flows in a pipe with ring‐type constrictions

The effects of Reynolds number on the physiological‐type of laminar pulsatile flow fields within the vicinity of mechanical ring‐type constriction in small pipes were studied numerically. The parameters considered are: the Reynolds number (Re) in the range of 50–1500; Strouhal number (St) in the range of 0.00156–3.98; Womersley number (Nw) from 0.0 to 50.0. The pulsatile flows considered were physiological‐type of simulated flows. Within a pulsating cycle, detailed flow characteristics were studied through the pulsating contours of streamline (ψ), vorticity (Ω), shear stress (τ) and isobar. The relations between the instantaneous flow rate (Q) and instantaneous pressure gradients (dp/dz) are observed to be elliptic. The relations between the instantaneous flow rate (Q) and pressure loss (P_loss) are quadratic. Linear relations were observed between the instantaneous flow rate (Q) and the maximum velocity, maximum vorticity and maximum shear stress. The Reynolds number of the flow in a pulsating cycle was found to have significant effects on the recirculation length and the pressure gradient within the pulsatile flow regime. Copyright © 1999 John Wiley & Sons, Ltd.     

The prediction of spatially periodic flows using a finite‐volume model

A periodic boundary condition has been developed that can be used in conjunction with a specified flow rate to produce accurate results in spatially periodic geometries. This condition is useful in situations where the flow rate is known, or more importantly, in cases where the pressure gradient is not known a priori, such as in countercurrent flows. Using the present condition, the flow rate is imposed at the inlet in terms of a bulk velocity, but the velocity field evolves as part of the solution. The condition is formulated to be suitable for both fixed and moving periodic domains. For the case of a moving domain, a correction is introduced to account for changes in the instantaneous velocity through the periodic edges. Under periodic conditions, these corrections integrate to zero over a complete (temporal) period. The new periodic condition is shown to produce accurate results for flat and wavy‐walled channels under both induced flow and countercurrent conditions. Copyright © 2003 John Wiley & Sons, Ltd.     

VORTEX SHEDDING RESONANCE FROM A ROTATIONALLY OSCILLATING CYLINDER

Vortex shedding resonance of a circular cylinder wake to a forced rotational oscillation has been investigated experimentally by measuring the velocity fluctuations in the wake, pressure distributions over the cylinder surface, and visualizing the flow field with respect to cylinder oscillations. The vortex shedding resonance occurs near the natural shedding frequency at small amplitude of cylinder oscillations, while the peak resonance frequency shifts to a lower value with an increase in oscillation amplitude. The drag and lift forces acting on the cylinder at fixed forcing Strouhal number indicate that the phase lag of fluid forces to the cylinder oscillations increases with an increase in oscillation amplitude, supporting the variation of resonance frequency with oscillation amplitude. The comparative study of the measured pressure distributions and the simultaneous flow visualizations with respect to cylinder rotation shows the mechanisms of phase lag, which is due to the strengthened vortex formation and the modification of the surface pressure distributions.     

An iterative method for the computation of the response of linearised Navier–Stokes equations to harmonic forcing and application to forced cylinder wakes

The central aim of this paper is the development and application of an efficient, iterative methodology for the computation of the perturbation fields induced by harmonic forcing of the linearised Navier–Stokes equations. The problem is formulated directly in the frequency domain, and the resulting system of equations is solved iteratively until convergence. The method is easily implemented to any implicit code that can solve iteratively the steady‐state Navier–Stokes equations. In this paper, it is applied to investigate the flow around a static cylinder with pulsating approaching flow and a cylinder undergoing forced stream‐wise oscillations. All terms of the perturbation kinetic energy equation are computed, and it is shown that perturbations grow by extracting energy from two sources: the underlying base flow field and the externally provided energy that maintains the imposed oscillation. The periodic drag force acting on the cylinder is also computed, and it is demonstrated that Morrison's equation is a simple model that can estimate with good accuracy the amplitude and phase of this force with respect to the approaching flow. The perturbation fields induced by periodic inlet flow (static cylinder) and forced stream‐wise cylinder oscillation are closely related: the velocity fields are identical in the appropriate reference frames, and a simple expression is derived, which links the pressures in the two flow cases. Copyright © 2014 John Wiley & Sons, Ltd.     

Unsteady flow in a circular sector duct

Analytic solutions for the unsteady flow in a circular sector duct are found using series sums of Bessel integrals. For starting flow due to a step pressure gradient, the velocity profile is initially flat, then approaches the rounded steady state shape in a time scale proportional to the square of opening angle of the sector. For oscillatory flow, the velocity is quasi-steady for low frequencies, but shows “annular effect” at large frequencies. Increased opening angle increases the amplitude and the phase lag. In all cases, the shear stress at the apex is zero for acute sector angles but becomes infinite for obtuse sector angles.     

Resistance and inertia of the flow of liquids in a tube or open canal

Summary When investigating the influence of an acceleration upon the velocity distribution and upon the resistance in a tube or canal, a distinction can be made between slowly varying motions where the resistance dominates, and quickly varying motions where the inertia dominates. When the motion varies quickly, practically all the liquid moves bodily, and the resistance only affects a small region near the walls. When the motion varies slowly, the velocity distribution differs from that of steady flow in that there is a phase lag of the central layers with respect to the peripheral layers.The following special types of motion were studied for laminar flow: forced oscillations in a round tube; forced oscillations in a crevice; the starting motion by a constant drop of potential in a round tube; free oscillations in a round tube. For free oscillations in a U-tube, the theory is checked by comparison with experiments (table I). General slow motion was studied in the following cases: laminar flow in tubes of elliptical (circular) and rectangular (square) cross-section, or in open canals of rectangular cross-section; fully developed turbulent flow in round tubes and wide open canals.The relation between drop of potential and total flow can be represented by the impedance of the tube or canal. For quick laminar motions the impedance is given by (52) and (53) and characterized by the high frequency inertanceH. For slow laminar motions the impedance is given by (49) and (50) and characterized by the resistanceR and the low frequency inertanceL. For slow turbulent motions the relation between drop of potential and total flow is given by (73). The low frequency inertance is always greater than the high frequency inertance. The difference represents the change in the resistance caused by the influence of the acceleration upon the distribution of the velocity. This difference betweenL andH amounts to 33% for laminar flow in a round tube, 38% in a square tube, and 20% in a crevice or a wide open canal. For turbulent flow it ranges from 1 to 8% in a rough round tube, from 1/2 to 3% in a smooth round tube, from 1/2 to 3 1/2% in a rough open canal or crevice, and from 1/4 to 1 1/2% in a smooth open canal or crevice (table III). For laminar flow the ratioL: H is equal to the ratio of the mean square and the square mean velocity in a steady flow.     

Combustion of premixed fuel and air downstream of a plane sudden-expansion

 Experiments have been performed to quantify the isothermal and combusting flows downstream of a plane sudden-expansion. The detailed measurements correspond to an area expansion ratio of 2.86 and a Reynolds number of 20000, and the combusting flows comprised premixed methane and air over a range of equivalence ratios with emphasis on values of 0.72 and 0.92 which gave rise to smooth and rough combustion, respectively. The results show that the extent of asymmetry of the isothermal flows was reduced by coupling the pressures between the two recirculation regions, and by imposing oscillations at the half-wave or full-wave frequency of the duct, and by combustion. Periodic variations of flame shape, velocities, acceleration, and temperature were observed in sympathy with the dominant pressure oscillation of rough combustion, and the length of the recirculation zones varied from less than 0.5 to 3 step heights. Rich and lean limits were established for combustion within the duct and, whereas the flame blew off at the lean limit, it detached from the expansion at the rich limit and stabilised on the flange at the duct exit. Within these limits, there were ranges of equivalence ratios over which the flame stabilised on one of the two steps with incomplete combustion. The imposition of oscillations narrowed the range of equivalence ratios over which the flame could be stabilised but reduced the equivalence ratio of the lean limit at which the flame could be stabilised on both steps and the effect increased with amplitude and was greatest when the frequency of the imposed oscillations corresponded to that of the half-wave in the duct. An increase in the amplitude of flow oscillations, natural or imposed, caused the concentrations of NO _x measured at the duct exit to decrease. Active control of flows with high amplitude of oscillations produced the expected reductions, but not over the entire measured range of equivalence ratio, and the imposition of pressure oscillations at the second harmonic of the half-wave frequency had a greater effect and over a wider range. Received: 14 September 1998 / Accepted: 19 February 1999     

Experimental Characterization of Premixed Flame Instabilities of a Model Gas Turbine Burner

In recent years, the NO _x emissions of heavy duty gas turbine burners have been significantly reduced by introducing premixed combustion. These highly premixed burners are known to be prone to combustion oscillations. In this paper, investigations of a single model gas turbine burner are reported focusing on thermo-acoustic instabilities and their interaction with the periodic fluctuations of the velocity and pressure. Phase-locked optical measurement techniques such as LDA and LIF gave insight into the mechanisms.Detailed investigations of a gas turbine combustor rig revealed that the combustor as well as the air plenum oscillate in Helmholtz modes. These instabilities could be attributed to the phase lag of the pressure oscillations between the air plenum and the combustor, which causes an acceleration and deceleration of the air flow through the burner and, therefore, alternating patterns of fuel rich and lean bubbles. When these bubbles reach the reaction zone, density fluctuations are generated which in turn lead to velocity fluctuations and, hence, keep up the pressure oscillations.With increasing the equivalence ratio strong combustion oscillations could be identified at the same frequency. Similarly as with weak oscillations, Helmholtz mode pressure fluctuations are present but the resulting velocity fluctuations in the combustor can be described as a pumping motion of the flow. By the velocity fluctuations the swirl stabilization of the flame is disturbed. At the same time, the oscillating pressure inside the combustor reaches its minimum value. Shortly after the flame expands again, the pressure increases inside the combustor. This phenomenon which is triggered by the pressure oscillations inside the air plenum seems to be the basic mechanism of the flame instability and leads to a significant increase of the pressure amplitudes.     

Numerical solution of oscillatory flow of Maxwell fluid in a rectangular straight duct

A numerical analysis is presented for the oscillatory flow of Maxwell fluid in a rectangular straight duct subjected to a simple harmonic periodic pressure gradient.The numerical solutions are obtained by a finite difference scheme method. The stability of this finite difference scheme method is discussed. The distributions of the velocity and phase difference are given numerically and graphically. The effects of the Reynolds number, relaxation time, and aspect ratio of the cross section on the oscillatory flow are investigated. The results show that when the relaxation time of the Maxwell model and the Reynolds number increase, the resonance phenomena for the distributions of the velocity and phase difference enhance.     

Oscillations of a viscous fluid in a thin layer between two coaxial disks rotating together

A study is made of the problem of the motion of an incompressible viscous fluid in the space between two coaxial disks rotating together with constant angular velocity under the assumption that the pressure changes in time in accordance with a harmonic law. The problem is solved using the equations of unsteady motion of an incompressible viscous fluid in a thin layer. It is shown that the velocity field in this case is a superposition on a steady field of damped oscillations with cyclic frequency equal to twice the angular velocity of the disks and forced oscillations with cyclic frequency equal to the cyclic frequency of the oscillations of the pressure field. It is shown that the amplitude of the forced oscillations of the velocity field depends strongly on the ratio of the cyclic frequency of the oscillations of the pressure field to the angular velocity of the disks. It is shown that there is a certain value of the ratio at which the amplitude of the forced oscillations has a maximal value (resonance). It is shown that even for very small amplitudes of the pressure oscillations the amplitude of the oscillations of the relative velocity at resonance may reach values comparable with the mean velocity of the main flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 166–169, January–February, 1984.     

Analysis of the flow in a thermo-acoustic MHD generator with conducting walls

This paper is devoted to the analysis of a pulsating MHD flow in a rectangular channel, submitted to an applied DC magnetic field and an imposed harmonically oscillating thermo-acoustically generated pressure gradient. The resulting AC current is collected by two electrodes placed on one either side of the channel. The walls of the channel are assumed electrically conducting. The channel is considered sufficiently long to justify the hypothesis of a fully developed flow. The analytical solution proposed is based on a simultaneous resolution of the Navier–Stokes and induction equations. This solution is limited to moderate values of the magnetic Reynolds number.     

Measurements of pulsating and oscillating flows of non-Newtonian fluids through concentric and eccentric cylinders

The unsteady flow of non-Newtonian fluids through concentric and eccentric cylinders was investigated experimentally. Two experiments were carried out; one was pulsating flow and the other was flow under a constant pressure gradient with the inner cylinder oscillating longitudinally. The flow enhancement was examined and its dependence on the frequency of the oscillations and the eccentricity of the apparatus was determined.     

Bifurcation and stability analysis of laminar flow in curved ducts

The development of viscous flow in a curved duct under variation of the axial pressure gradient q is studied. We confine ourselves to two‐dimensional solutions of the Dean problem. Bifurcation diagrams are calculated for rectangular and elliptic cross sections of the duct. We detect a new branch of asymmetric solutions for the case of a rectangular cross section. Furthermore we compute paths of quadratic turning points and symmetry breaking bifurcation points under variation of the aspect ratio γ (γ=0.8…1.5). The computed diagrams extend the results presented by other authors. We succeed in finding two origins of the Hopf bifurcation. Making use of the Cayley transformation, we determine the stability of stationary laminar solutions in the case of a quadratic cross section. All the calculations were performed on a parallel computer with 32×32 processors. Copyright © 2009 John Wiley & Sons, Ltd.     

Numerical predictions of flow and thermal fields in an enclosure with two periodically moving walls

Numerical predictions of transient flow and thermal fields in a rectangular enclosure with two periodically moving vertical walls are presented. The combined influence of the movement of the walls and the buoyancy as well on the flow pattern and heat transfer performance is evaluated. The compressible‐flow model is adopted, and governing equations are expressed in integral form and discretized on the moving grids, which deform in resonance with the walls to accommodate the variation in the volume of the enclosure. A two‐stage pressure‐correction scheme is applied for simultaneously determining the distributions of absolute pressure, density, temperature, and velocity of the compressible flow field in the enclosure during the periodically stable periods. Effects of the frequency, stroke, and the phase angle of the wall oscillations on the flow are of major concerns in this study. The frequency is ranged between 5 and 25 Hz and the dimensionless strokes (l/H) of the wall are varied from 0.4 to 1.0. Results for Nusselt numbers on the walls as well as the dimensionless input work required to excite the wall oscillation are provided. Copyright © 2006 John Wiley & Sons, Ltd.     

A numerical study of pulsatile laminar flows in a pipe with a ring-type constriction

Numerical simulations have been carried out to study pulsatile laminar flows in a pipe with an axisymmetric ringtype constriction. Three types of pulsatile flows were investigated, namely a physiological flow, a pure sinusoidal flow and a non-zero mean velocity sinusoidal flow. The laminar flow governing equations were solved by the SIMPLE algorithm on a non-staggered grid and a modified Crank-Nicolson approximation was used to discretrize the momentum equations with respect to time. The maximum flow Reynolds numer (Re) is 100. The Womersley number (N_w) ranges from 0 to 50, with the corresponding Strouhal number (St) ranging from 0 to 3·98. The constriction opening ratio (d/D) and thickness ratio (h/D) are fixed at 0·5 and 0·1 respectively. Within the time period investigated, all these pulsatile flows include both forward and backward flows. The unsteady recirculation region and the recirculation points change in size and location with time. For N_w ≤ 1 and St≤ 1·56 x 10^?3 the three pulsatile flows have the same simple relation between the instantaneous flow rate and pressure loss (Δp) across the constriction and the pressure gradient in the axial direction (dp/dz) in the fully developed flow region. The phase angles between the flow rate and pressure loss and the pressure gradient are equal to zero. With increasing N_w and St, the phase angle between the flow rate and the dp/dz becomes larger and has its maximum value of 90° at N_w = 50 and St = 3·98. The three pulsatile flows also show different relations between the flow rate and the pressure gradient. The pure sinusoidal flow has the largest maximum pressure gradient and the non-zero mean velocity sinusoidal flow has the smallest. For larger N_w and St the fully developed velocity profiles in the fully developed flow region have a smaller velocity gradient along the radial direction in the central region. The maximum recirculation length increases for N_w ranging from 0 to 4·2, while this length becomes very small at N_w = 50 and St = 3·98. The deceleration tends to enlarge the recirculation region and this effect appears for N_w ≥ 3 and St ≥ 1·43×10^?2. Linear relations exist between the flow rate and the instantaneous maximum values of velocity, vorticity and shear stress.     

A convective weakly viscoelastic rotating flow with pressure Neumann condition

The objective of this work is to investigate through the numeric simulation, the effects of the weakly viscoelastic flow within a rotating rectangular duct subject to a buoyancy force due to the heating of one of the walls of the duct. A direct velocity–pressure algorithm in primitive variables with a Neumann condition for the pressure is employed. The spatial discretization is made with finite central differences on a staggered grid. The pressure field is directly updated without any iteration. Numerical simulations were done for several Weissemberg numbers (We) and Grashof numbers (Gr) . The numerical results show that for high Weissemberg numbers (We>7.4 × 10^?5) and for ducts with aspect ratio 2:1 and 8:1, the secondary flow is restabilized with a stretched double vortex configuration. It is also observed that when the Grashof number is increased (Gr>17 × 10^?4) , the buoyancy force neutralizes the effects of the Coriolis force for ducts with aspect ratio 8:1. Copyright © 2008 John Wiley & Sons, Ltd.